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The Fundamental Principles of Composite Material Stiffness Predictions David Richardson Contents • Description of example material for analysis • Prediction of Stiffness using… – Rule of Mixtures (ROM) – ROM with Efficiency Factor – Hart Smith 10% rule – Classical Laminate Analysis • Simplified approach • Overview of misconceptions in material property comparison between isotropic materials and composites Lamina Axis Notation Diagram taken from Harris (1999) Example Material for Analysis • M21/35%/UD268/T700 – A common Aerospace uni-directional pre-preg material called HexPly M21 from Hexcel • Ef = 235 GPa Em= 3.5 GPa • ρf = 1.78 g/cm3 ρm = 1.28 g/cm3 • Wr = 35% (composite resin weight fraction) • Layup = (0/0/0/+45/-45/0/0/0) Stage 1 • Convert fibre weight fraction of composite to fibre volume fraction – Fibre weight fraction used by material suppliers – Fibre volume fraction needed for calculations Fibre Volume Fraction • Fibre mass fraction of M21 = 65% (0.65) – Data sheet says material is 35% resin by weight, therefore 65% fibre by weight • Calculation of fibre volume fraction • The resulting volume fraction is 57.2% Wf f Vf Wf Wm f m Methods of Stiffness Prediction • Rule of Mixtures (with efficiency factor) • Hart-Smith 10% Rule – Used in aerospace industry as a quick method of estimating stiffness • Empirical Formulae – Based solely on test data • Classical Laminate Analysis – LAP software Rule of Mixtures • A composite is a mixture or combination of two (or more) materials • The Rule of Mixtures formula can be used to calculate / predict… – Young’s Modulus (E) – Density – Poisson’s ratio – Strength (UTS) • very optimistic prediction • 50% usually measured in test • Strength very difficult to predict – numerous reasons Rule of Mixtures for Stiffness • Rule of Mixtures for Young’s Modulus • Assumes uni-directional fibres • Predicts Young’s Modulus in fibre direction only • Ec = EfVf + EmVm • Ec = 235×0.572 + 3.5×0.428 • Ec = 136 GPa Rule of Mixtures: Efficiency Factor • The Efficiency Factor or Krenchel factor can be used to predict the effect of fibre orientation on stiffness • This is a term that is used to factor the Rule of Mixtures formula according to the fibre angle – See following slide Reinforcing Efficiency an= proportion of total fibre content 𝜃 = angle of fibres 𝜂𝜃= composite efficiency factor (Krenchel) Efficiency (Krenchel) Factor Diagram taken from Harris (1999) Prediction of E for Example Ply Ef = 235 GPa Em= 3.5 GPa Vf = 0.572 E(θ) = (Cos4θ × 235×0.572) + (3.5×0.428) Predicted modulus versus angle plotted on following slide Prediction of Tensile Modulus (Efficiency Factor) 140 120 Tensile Modulus (GPa) 100 80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 Angle (degrees) Efficiency Factor for Laminate • Layup = (0/0/0/+45/-45/0/0/0) • η = Cos4θ • 0° = η = 1 • 45° = η = 0.25 • 90° = η = 0 • Laminate in X-direction • (6/8 × 1) + (2/8 × 0.25) • (0.75 + 0.0625) • 0.8125 • Laminate in Y-direction • (6/8 × 0) + (2/8 × 0.25) • (0 + 0.0625) • 0.0625 Prediction of E for Example Ply Ef = 235 GPa Em= 3.5 GPa Vf = 0.572 Ex = (0.8125 × 235×0.572) + (3.5×0.428) Ex = 109 + 1.5 = 110.5 GPa Ey = (0.0625 × 235×0.572) + (3.5×0.428) Ey = 8.4 + 1.5 = 9.9 GPa Ten-Percent Rule • Hart-Smith 1993 – Each 45° or 90° ply is considered to contribute one tenth of the strength or stiffness of a 0° ply to the overall performance of the laminate – Rapid and reasonably accurate estimate – Used in Aerospace industry where standard layup [0/±45/90] is usually used Ex = E11 . (0.1 + 0.9 × % plies at 0°) σx = σ11 . (0.1 + 0.9 × % plies at 0°) Gxy = E11 . (0.028 + 0.234 × % plies at ± 45°) Prediction of Tensile Modulus 140 120 ROM 100 Tensile Modulus (GPa) Hart-Smith 80 60

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